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sinpi(2x-5)/2=-sqrt2/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                        ___ 
sin(p)*I*(2*x - 5)   -\/ 2  
------------------ = -------
        2               2
isin⁡(p)(2x−5)2=(−1)22\frac{i \sin{\left(p \right)} \left(2 x - 5\right)}{2} = \frac{\left(-1\right) \sqrt{2}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
isin⁡(p)(2x−5)2=(−1)22\frac{i \sin{\left(p \right)} \left(2 x - 5\right)}{2} = \frac{\left(-1\right) \sqrt{2}}{2}
cambiamos
i(2x−5)sin⁡(p)2−1+22=0\frac{i \left(2 x - 5\right) \sin{\left(p \right)}}{2} - 1 + \frac{\sqrt{2}}{2} = 0
isin⁡(p)(2x−5)2−1−(−1)22=0\frac{i \sin{\left(p \right)} \left(2 x - 5\right)}{2} - 1 - \frac{\left(-1\right) \sqrt{2}}{2} = 0
Sustituimos
w=sin⁡(p)w = \sin{\left(p \right)}
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 - -sqrt+2)/2 + i*w2*x/2+5/2 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 + sqrt(2)/2 + i*w*(-5 + 2*x)/2 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
iw(2x−5)2+22=1\frac{i w \left(2 x - 5\right)}{2} + \frac{\sqrt{2}}{2} = 1
Dividamos ambos miembros de la ecuación en (sqrt(2)/2 + i*w*(-5 + 2*x)/2)/w
w = 1 / ((sqrt(2)/2 + i*w*(-5 + 2*x)/2)/w)

Obtenemos la respuesta: w = i*(-2 + sqrt(2))/(-5 + 2*x)
hacemos cambio inverso
sin⁡(p)=w\sin{\left(p \right)} = w
sustituimos w:
Gráfica
Respuesta rápida [src] 
                         ___                                                        ___                                   
     5                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
x1 = - + ------------------------------------------------------- + -------------------------------------------------------
     2     /   2            2              2           2       \     /   2            2              2           2       \
         2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
x1=52+2isin⁡(re⁡(p))cosh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))x_{1} = \frac{5}{2} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
x1 = 5/2 + sqrt(2)*i*sin(re(p))*cosh(im(p))/(2*(sin(re(p))^2*cosh(im(p))^2 + cos(re(p))^2*sinh(im(p))^2)) + sqrt(2)*cos(re(p))*sinh(im(p))/(2*(sin(re(p))^2*cosh(im(p))^2 + cos(re(p))^2*sinh(im(p))^2))
Suma y producto de raíces [src] 
suma
                    ___                                                        ___                                   
5                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- + ------------------------------------------------------- + -------------------------------------------------------
2     /   2            2              2           2       \     /   2            2              2           2       \
    2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
52+2isin⁡(re⁡(p))cosh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))\frac{5}{2} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
=
                    ___                                                        ___                                   
5                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- + ------------------------------------------------------- + -------------------------------------------------------
2     /   2            2              2           2       \     /   2            2              2           2       \
    2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
52+2isin⁡(re⁡(p))cosh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))\frac{5}{2} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
producto
                    ___                                                        ___                                   
5                 \/ 2 *cos(re(p))*sinh(im(p))                             I*\/ 2 *cosh(im(p))*sin(re(p))            
- + ------------------------------------------------------- + -------------------------------------------------------
2     /   2            2              2           2       \     /   2            2              2           2       \
    2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/   2*\cos (re(p))*sinh (im(p)) + cosh (im(p))*sin (re(p))/
52+2isin⁡(re⁡(p))cosh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))+2cos⁡(re⁡(p))sinh⁡(im⁡(p))2(sin⁡2(re⁡(p))cosh⁡2(im⁡(p))+cos⁡2(re⁡(p))sinh⁡2(im⁡(p)))\frac{5}{2} + \frac{\sqrt{2} i \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} + \frac{\sqrt{2} \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(p\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(p\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
=
  /  ___                           \                           
I*\\/ 2  + 5*cos(re(p))*sinh(im(p))/ + 5*cosh(im(p))*sin(re(p))
---------------------------------------------------------------
     2*(cosh(im(p))*sin(re(p)) + I*cos(re(p))*sinh(im(p)))
i(5cos⁡(re⁡(p))sinh⁡(im⁡(p))+2)+5sin⁡(re⁡(p))cosh⁡(im⁡(p))2(sin⁡(re⁡(p))cosh⁡(im⁡(p))+icos⁡(re⁡(p))sinh⁡(im⁡(p)))\frac{i \left(5 \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)} + \sqrt{2}\right) + 5 \sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)}}{2 \left(\sin{\left(\operatorname{re}{\left(p\right)} \right)} \cosh{\left(\operatorname{im}{\left(p\right)} \right)} + i \cos{\left(\operatorname{re}{\left(p\right)} \right)} \sinh{\left(\operatorname{im}{\left(p\right)} \right)}\right)} 
(i*(sqrt(2) + 5*cos(re(p))*sinh(im(p))) + 5*cosh(im(p))*sin(re(p)))/(2*(cosh(im(p))*sin(re(p)) + i*cos(re(p))*sinh(im(p))))
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